Question

A cylinder with the greatest lateral surface is inscribed in a cone. Find the ratio of the altitude of the cone to that of the cylinder.

Answer 1

Let radius and height of cone be R and H respectively and r and h be the radius and height of cylinder

Then, $S=2\pi rh$

Now, $h=-\frac{H}{R}r+H$

$\Rightarrow S=2\pi r[-\frac{H}{R}r+H]=2\pi rH-\frac{2\pi H}{R}{r}^2$

$\frac{dS}{dr}=0\Rightarrow 2\pi H-\frac{4\pi H}{R}r=0\Rightarrow R-2r=0\Rightarrow r=\frac{R}{2}$

$\therefore h=\frac{-H}{R}\times \frac{R}{2}+H\Rightarrow h=\frac{H}{2}\Rightarrow \frac{H}{h}=2$